REPORT ON THE RADIOLARIA. Xlll 



been incorrectly called " spherical" belong to this category, for they are none of them 

 true spheres in the geometrical sense (like the central capsules of the Spheeroidea), 

 but rather endospherical polyhedra, whose angles are indicated by the nodal points of 

 the lattice-shell, or the radial spines which spring from them. These endospherical 

 polyhedra may be divided into three groups, the regular, subregular, and irregular. Of 

 regular polyhedra, properly so-called, it may be shown geometrically that only five can 

 exist, namely, the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 

 All these are actually manifested among the Radiolaria, although but seldom. Much 

 more common are the subregular endospherical polyhedra, e.g., spherical lattice-shells 

 with regular hexagonal meshes of equal size ; they are never exactly equal nor perfectly 

 regular, but the divergences are so insignificant that they escape superficial observation 

 (PL 20, figs. 3, 4 ; PL 26, figs. 1-3). On the contrary in the irregular endospherical 

 polyhedra the meshes of the lattice-sphere are more or less different in size and often in 

 form also (PL 28, figs. 4, 8 ; PL 30, figs. 4, 6). The five truly regular polyhedra require 

 separate notice on account of their importance. (See Gener. MorphoL, Bd. i. p. 406.) 



26. The Regular Icosahedral Ground-Form. The ground-form whose geometrical 

 type is the regular icosahedron (bounded by twenty equilateral triangles) is rarely 

 exemplified, but it occurs among the PH.EODARIA in the Circoporid genus Circogonia 

 (PL 117, fig. 1), and also in certain Aulosphaerida, but, apparently, only as an 

 accidental variation (e.g., Aulosphcera icosahedra}. Furthermore, this ground-form may 

 also be assumed to occur in those Sphseroidea whose spherical lattice-shells bear 

 twelve equidistant radial spines (e.g., many species of Acanthosphcera, Heliosphcera, and 

 other Astrosphserida) ; the basal points of these spines indicate the twelve angles of the 

 regular icosahedron. (See on this head Gener. MorphoL, Bd. i. p. 411.) 



27. The Regular Dodecahedral Ground-Form. The ground-form whose geometrical 

 type is the regular dodecahedron (or pentagonal dodecahedron), bounded by twelve 

 equilateral and equiangular pentagons, is very rarely found perfectly developed, as 

 in Circorrhegma dodecahedra (PL 117, fig. 2). This form is by no means so common 

 among the Radiolaria as in the pollen grains of plants (e.g., Buchholzia maritima, 

 Fumaria spicata, Polygonum amphibium, &c.). It can, however, be regarded as present 

 in all those Sphseroidea whose spherical lattice-shells bear twenty equal and 

 equidistant radial spines (e.g., many species of Acanthosphcera, Heliosphcera, and other 

 Astrosphaerida) ; the basal points of these spines mark out the twenty angles of the 

 regular pentagonal dodecahedron. (See Gener. MorphoL, Bd. i. p. 412.) 



28. The Regular Octahedral Ground-Form. The ground-form whose geometrical 

 type is the regular octahedron (bounded by eight equilateral triangles), commonly 

 appears among the SPUMELLARIA in the family Cubosphserida (p. 169, Pis. 21-25). In 



